A circle has a radius of $3$. An arc in this circle has a central angle of $240^\circ$. What is the length of the arc? ${6\pi}$ ${240^\circ}$ $\color{#DF0030}{4\pi}$ ${3}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (3) = 6\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{240^\circ}{360^\circ} = \dfrac{s}{6\pi}$ $\dfrac{2}{3} = \dfrac{s}{6\pi}$ $\dfrac{2}{3} \times 6\pi = s$ $4\pi = s$